I am looking for the expression of the expected value of the max $M$ i.i.d. Gumbel $\epsilon_i$ with scale $\beta$ and location $\mu$. I found the answer to this question when $\beta=1$ (e.g., here). I am looking for the case in which $\beta\neq 1$. When $\beta=1$ $$ \mathbb{E}(\max_{i\in \{1,...,M\}} \epsilon_i+ a_i) = \mu+\gamma+\log[\sum_{i=1}^M \exp(a_i+\mu)] $$ where $\gamma$ is the Euler-Mascheroni constant.
Could you confirm that when $\beta\neq 1$ we get $$ \mathbb{E}(\max_{i\in \{1,...,M\}} \epsilon_i+ a_i) = \mu+\beta\gamma+\beta\log[\sum_{i=1}^M \exp(\frac{a_i+\mu}{\beta})] $$ ?